Abu Rayhan al-Biruni · Science
Al-Biruni’s celebrated dip-angle method at the fort of Nandana: with one hill of known height, an astrolabe, and a little trigonometry, a single observer standing in one place computed the radius of the whole earth - and measurement, for him, was the beating heart of science.
How could anyone, a thousand years ago, measure the size of the whole earth? The classic answer was heroic and costly. Eratosthenes had compared the sun’s shadow at two cities a known distance apart; the astronomers of the caliph al-Ma’mun, two centuries before al-Biruni, had sent survey parties to pace out a degree of the meridian across the flat plain of Sinjar. Both methods needed two widely separated places, a carefully measured baseline, and a small army of surveyors trudging across the ground. Al-Biruni, sometime around 1018 to 1025 while he was in the Punjab in the train of Mahmud of Ghazna, found a way to do it from one spot. The idea turns on a fact any sailor knows: from a height, the line to the horizon tips slightly downward, because the earth curves away beneath you. That tiny ‘dip’ angle secretly encodes the curvature of the entire planet. Read it precisely, and the earth gives up its radius.
Al-Biruni carried out the measurement at the fort of Nandana, in the Salt Range of what is now Pakistan, where a mountain rose above a wide, flat plain - the perfect stage. First he measured the perpendicular height of the hill (he reports a value of about 652 cubits, roughly 300 metres) using a careful survey of the slope. Then he waited, as he tells us, for a day when the air was clear, climbed to the summit, and with a graduated instrument sighted the far line where the earth seemed to meet the sky. He found the horizon dipped below the horizontal by 0 degrees 34 minutes - about half a degree, a quantity so small that the whole result hangs on measuring it well. From the height h and the dip α, the radius of the earth follows from a single relation of the right triangle: the radius equals h times the cosine of α, divided by one minus that cosine. Al-Biruni turned the crank and out came the size of the world.
The mountain at Nandana is a window onto the whole cast of al-Biruni’s mind. He was, above everything, a measurer. Across his life he determined latitudes and longitudes for hundreds of towns, timed eclipses to fix the distance between cities, weighed metals and gemstones to fractions that rival modern values, and computed the earth’s size from a hilltop. For him, to know a thing scientifically was to put a number on it, and to say honestly how good that number was. He records his instruments, his conditions (‘when the air was clear’), his sources of error, and the arithmetic - so that a reader can check him. This is the ethic that runs through all five of these lessons: not to accept a claim because an authority made it, but to go out, observe, measure, and compare. The earth’s radius from a single mountain is the emblem of that ethic, one angle standing in for a planet.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that at Nandana al-Biruni measured the height of a hill and the small angle by which the horizon dips when seen from its summit, then used trigonometry to compute the earth’s radius from a single spot. Explain how the method works, and why the accuracy of his result depends on how long his ‘cubit’ actually…
Leads to Eratosthenes.
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